# How do you identity if the equation 9x^2+4y^2-36=0 is a parabola, circle, ellipse, or hyperbola and how do you graph it?

Jan 9, 2017

This is the equation of an ellipse.
See the graph below

#### Explanation:

We can do the discriminant test

$A {x}^{2} + B x y + C {y}^{2} + D x + E y + F = 0$

Discriminant, $\Delta = {B}^{2} - 4 A C$

Here,

$9 {x}^{2} + 4 {y}^{2} - 36 = 0$

$\Delta = 0 - 4 \cdot 9 \cdot 4 = - 144$

As $\Delta < 0$, we conclude that the equation represents an ellipse.

Let's rearrange the equation

$9 {x}^{2} + 4 {y}^{2} = 36$

Divide throughly by $36$

$\left(9 {x}^{2} / 36\right) + \left(4 {y}^{2} / 36\right) = \frac{36}{36}$

${x}^{2} / 4 + {y}^{2} / 9 = 1$

We can compare this to

${x}^{2} / {a}^{2} + {y}^{2} / {b}^{2} = 1$

$a = 2$ and $b = 3$

The vertices are $\left(2 , 0\right)$, $\left(- 2 , 0\right)$, $\left(0 , 3\right)$. and $\left(0 , - 3\right)$

graph{(x^2/4)+(y^2/9)=1 [-8.89, 8.89, -4.444, 4.44]}